Question

Let z and $\omega$ be two nonzero complex numbers such that $\left|z\right|=\left|\omega \right|$ and $argz=\pi -arg\omega$, then z equals

• A. $\omega$
• B. ${}^{-}\omega$
• C. $\overline{\omega }$
• D. ${}^{-}\overline{\omega }$

Answer 1

Answer: (D)

${}^{-}\overline{\omega }$

$\left|z\right|=\left|w\right|argz=\pi -argw$
Let $z=r(\cos A+i\sin A)\&w=r(\cos B+i\sin B)$
Since, $argz=\pi -argw$
$\Rightarrow A=\pi -B$
$z=r(\cos (\pi -B)+i\sin (\pi -B))=r(-\cos B+i\sin B)=-r(\cos B-i\sin B)\Rightarrow z=-\overline{w}$
Ans: D